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Masters Theses Student Theses and Dissertations
1970
Linear analysis of pneumatic dashpot damping Linear analysis of pneumatic dashpot damping
Nathalal Gordhanbhai Patel
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Recommended Citation Recommended Citation Patel, Nathalal Gordhanbhai, "Linear analysis of pneumatic dashpot damping" (1970). Masters Theses. 7152. https://scholarsmine.mst.edu/masters_theses/7152
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LINEAR ANALYSIS OF PNEUMATIC
DASHPOT DAMPING
BY
NATHALAL GORDHANBHAI PATEL, 1942 -
A
THESIS
submit t ed to the faculty of
THE UNIVERSITY OF MISSOURI - ROLLA
in partial fulfi llment of the requirements for the
Degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
Rolla, Missouri
1970
Approved by
ABSTRACT
The linear analysis of pneumatic damping with orifice or
capillary restrictions and with or without mean flow is presented.
Temperature and pressure relationships are derived from energy
considerations rather than the usual simplification of an assumed
polytropic coefficient. Analytical results are compared with the
experimental results and a coefficient of heat transfer, Ch, is
defined for calculation of the equivalent linear damping factor Cd.
ii
ACKNOWLEDGEMENTS
The author wishes to extend his sincere thanks and appreciation
to Dr. D. A. Gyorog for the guidance, encouragement and valuable
suggestions throughout the course of this thesis.
The author is thankful to Mr. Dick Smith for his technical
assistance and to Elizabeth Wilkins for her cooperation in typing
the thesis.
lll
iv
TABLE OF CONTENTS
ABSTRACT .... ii
ACKNOWLEDGEMENT lll
LIST OF ILLUSTRATIONS vi
\ LIST OF TABLES . . viii
LIST OF SYMBOLS ix
I. INTRODUCTION 1
II. EXPERIMENTAL PROCEDURE 5
III. LINEAR ANALYSIS .... 10
A. Dead-ended Chamber 13
B. Single Orifice Restriction 17
C. Single Capillary Restriction 26
D. Two Orifices in Series with Mean Flow 29
E. Two Capillaries in Series with Mean Flow 34
IV. DISCUSSION 39
V. CONCLUSION 47
APPENDIX A System dimensions and instrumentation calibration . . . . . . . 48
A.1 Pneumatic dashpot dimensions 48
A.2 Displacement transducer calibration 48
A.3 Pressure transducer coefficients 48
A.4 Oscilloscope trace area measurement 48
A.5 Magnitude ratio and phase angle measurements 50
A.6 Uncertainties in experiments 52
APPENDIX B Derivation of equivalent heat transfer coefficient, Ch . . . . . . . . . . . . 55
Table of contents (continued)
APPENDIX C Experimental results ....
C.l Experimental data reduction
C.2 Time constant calculations .
C.3 Tables and typical test run photographs
BIBLIOGRAPHY
VITA ....
v
Page
60
60
60
63
76
77
Figure
1. Experimental test. instrumentation
vi
LIST OF ILLUSTRATIONS
Mechanical arrangement and 6
2. Illustration of the system model 10
3. The equivalent damping coefficient as a function of frequency 16
4.
5.
6.
7.
8.
9.
Non-dimensional log-log plot of the magnitude ratio
I~: I versus WT . . . . . . . . . . 0
Phase angle cj> versus non-dimensional term WT
cd Non-dimensional log-log plot of ~versus
p 0
single orifice restriction cd
Non-dimensional log-log plot of ~ versus p c
single capillary restriction
Non-dimensional log-log plot of k cd
versus T p 00
two orifices in series with mean flow . .
Non-dimensional log-log plot of k cd
versus T p cc
two capillaries in series with mean flow
. .
0 I ~: l w-r for a
0
w-r for a c
WT for 00
. .
WT for cc
.
. . . 20
. . 21
23
28
. . . . 33
. . . . 38
10. Magnitude ratio, phase angle and equivalent damping coefficient, Cd' as a function of frequency for different values of n . . . . . . . . . . . . . . . . . . . . . . . 41
11. (a) Piston cylinder arrangement (b) Bellows chamber arrangement
12. LVDT calibration curves
13. Formation of elliptical loop and measurements of magnitude
44
49
ratio and phase angle . . . . . . . . . . . . . 51
14. Equivalent heat transfer coefficient, Ch' as a function of frequency w rad/sec . . . . . . . . . . . . . 56
15. Log of equivalent heat transfer coefficient, Ch' versus log of Reynolds Number . . . . . . . . . . . . . . . . 58
Vll
Figure Page
16. Photograph of test run No: 2 of TABLE III 69
17. Photograph of test run No: 8 of TABLE III 69
18. Photograph of test run No: 2 of TABLE IV 70
19. Photograph of test run No: 5 of TABLE IV 70
20. Photograph of test run No: 9 of TABLE IV 71
21. Photograph of test run No: 10 of TABLE IV 71
22. Photograph of test run No: 2 of TABLE v 72
23. Photograph of test run No: 3 of TABLE v 72
24. Photograph of test run No: 5 of TABLE v 73
25. Photograph of test run No: 7 of TABLE v 73
26. Photograph of test run No: 3 of TABLE VI 74
27. Photograph of test run No: 8 of TABLE VI 74
28. Photograph of test run No: 3 of TABLE VII 75
29. Photograph of test run No: 7 of TABLE VII 75
viii
LIST OF TABLES
TABLE
I Test run conditions ..... . 8
II Cornp ari son of calculated values for the adiabatic process assumption . . 40
III Experimental data for dead-ended chamber 64
IV Experimental data for single orifice restriction 65
v Experimental data for single capillary restriction 66
VI Experimental data for two orifices in series with mean through- flow . . . . . . . . . . . . 67
VII Experimental data for two capillaries in series with mean through- flow . . . . . . . . . . . . . 68
A
A s
c p
c v
E
h
K
k p
k
L
LIST OF SYMBOLS
Effective orifice and capillary
restrictions
Cross-sectional area of the bellows
Surface area of the bellows
Constant used in capillary formula
Constant used in capillary formula
Equivalent damping coefficient
Specific heat at constant pressure
Specific heat constant volume
Equivalent heat transfer coefficient
Diameters
Mean diameter of the bellows
Energy
Coefficient of heat transfer
Constant in orifice flow formula
Pneumatic spring rate
Thermal conductivity
Equivalent length of the bellows
. 2 1n
. 2 1n
. 2 1n
in lb-sec
lb-sec in
in-lb
sec0
R
inches
inches
in-lb
lb -.-ln
Btu
inches
ix
L12' T
l'2 3 Lengths of capillary tubes
M20 Mass of air in the bellows chamber
m2 Variation in M20
. M12' M23 Mass flow rate of air
M120' M230 Steady-state mass flow rate
Variations of M12
, M23
n Polytropic process constant
P1
, P2
, P3
Absolute pressures
P10
, P20
, P30
Steady-state pressures
Q Rate of heat trans fer
q Variation of Q
R Gas constant
R e Reynolds number
T1
, T2
, T3
Absolute temperature
T10
, T20
, T30
Steady-state temperatures
t time
U Any variable, or internal energy
V Velocity
inches
lb m
lb m
lb m
sec
lb m
sec
lb m
sec
psi a
psi a
psi a
in-lb sec
in-lb sec
sec
in-lb
in sec
X
w
. w
X
X m
X
y
z
z
y
T
w
Mean volume of the bellows
Steady-state volume of the bellows
Variation of v2
Work
Variation of W
Stroke
Maximum value of X
Variation of X
Any variable
A general function
Variation of Z or height
Specific heat ratio
Time constant
Frequency
Phase angle
. 3 ~n
. 3 ~n
3 ln
in-lb sec
in-lb sec
in
ln
in
in
sec
rad/sec
deg
xi
I. INTRODUCTION
The dissipation of energy by damping devices is required in
many systems to improve their transient behavior. In many fluid
systems the dynamic instability of control valves can lead to large
oscillations, if sufficient damping is not provided. This damping
characteristic can be provided by any mechanism such as viscous
effects, heat transfer effects, etc., which create a phase dis
placement between the forces and the resulting displacements or
changes in volume. Electrodynamic damping which is employed widely
in instruments is based on the use of a coil capable of movement
within a stationary and uniform magnetic field. Hydraulic dashpots
with linear flow restrictions give rise to viscous damping.
Friction damping is also established when two surfaces slide over
each other without lubrication. A typical example of this coulomb
friction is the laminated spring normally used on vehicles.
Internal structural damping may result when elastic materials such as
metals are subjected to cyclic load reversal. On the stress strain
diagram, the area within the hysteresis loop is proportional to the
amount of energy dissipated. Metalic bars may also exhibit damping
due to thermal effects. If stress and deformation are related to
temperature and hence heat transfer it is seen that damping must
result.
In a dashpot (or bellows), under the action of a harmonic
disturbance, damping is generated by a phase difference between
the instantaneous volume and pressure. This is a direct result of
the amount of air flowing through the restriction and is called
1
pneumatic damping. In a linear system~ liquids and gases generally
display elliptical hysteretic loops of pressure versus displacement,
when any damping mechanism is involved. For example, Fig. 13,
Appendix A-5, the area within the loop formed is a measure of the
cyclic energy dissipated by damping. In general, this type of
damping is non-linear since the force produced may not be a linear
function of velocity. However, for small amplitudes a linear
analysis can be applied with a very close agreement to the actual
damping. This is fortunate since it provides a much more convenient
technique for preliminary design and analysis of pneumatic systems.
The equivalent damping coefficient, Cd, can be defined by equaling
the linear damping per cycle to the actual energy dissipated per
cycle.
2
One important damper, from a practical standpoint, is the air
filled dashpot. Pneumatic damping has been used to improve the
response of instruments such as microphones and linear accelerometers.
Optimum design of pneumatic system damping at any frequency can be
achieved by varying the size and geometry of the flow restriction
and the variable volume chamber. R. L. Reskin [1] assumed in his
analysis of pneumatic damping that mass transfer through a porous
plug used in his experiment was the only significant mechanism for
energy dissipation. The porous plug provides the direct control
over the time lag between the pressure and volume. Darcy's Law of
flow through the porous media was considered to analyze the damping
in the system and it was proved that damping was proportional to the
phase difference. However, for his work the stroke was on the order
3
of ± 0.0005 inches and his analysis assumed that the temperature
remained constant.
R. W. Townsend [2] has analyzed pneumatic damping for a single
orifice restriction. His analytical curve and data agree near the
breakpoint frequency on a log-log plot of non-dimensionalized
quantities of the equivalent damping coefficient versus frequency,
but the low and high frequency asymptotes do not seem to have proper
slopes. Damping effects due to flow through orifices and capillaries
were also analyzed by B. W. Andersen [ 3]. Both investigators assumed
a polytropic relation between pressure and temperature to account for
heat transfer effects in obtaining the equivalent damping coefficient.
The value of n is this polytropic relation of
can be assumed to be 1.0 for isothermal assumption, but for actual
non-reversible processes the value of n can be much greater or less
than 1.
The objective of this investigation is to verify the analytical
result of the equivalent damping coefficient with the experimental
data. An analysis relating continuity, energy and equation of state
relations to the frequency response of the system will be developed.
Cases for orifices and capillaries with mean flow and without mean
flow will be considered. An equivalent damping coefficient, Cd'
will be obtained from the work per cycle, which will be measured.
Equating this factor to the linear analysis result will provide the
equivalent heat transfer coefficient, ~· The calculated results
will be compared to the experimental data and also to Townsend's [2]
experimental results. From the analysis pneumatic damper design
criteria will be suggested.
4
II. EXPERIMENTAL PROCEDURE
A schematic drawing of the experimental apparatus and the
instrumentation are illustrated in Fig. 1. A permanent magnet type
!VIB vibration exciter, Model PM100, was driven by a power amplifier
Model 2250 MB (MB Electronic Company). The sinusoidal voltage input
signal was obtained with a Hewlett-Packard Company oscillator.
Maximum force for the shaker was rated at 100 lbs and the acceleration
was further limited to 100 g.
A Kistler piezoelectric pressure transducer, Model 6011, was
used to measure the changes in pressure between the bellows and
surroundings. The stroke of the shaker table was measured with a
linear variable differential transformer (L.V.D.T.). The amplitude
of the stroke was maintained constant as the frequency was varied
from 2 cps to 10 cps, by varying the power input to the shaker.
The mean bellows length, about which oscillations took place,
was selected as one-half of its allowable deflection in compression.
Therefore the bellows was operated within its elastic range for the
test stroke magnitudes. Three equally spaced springs were installed
around the external circumference of the bellows to prevent the
bellows from expanding for the flow tests since the mean air pressure
within the bellows was greater than atmospheric for these tests.
Bolts were attached to both ends of the springs so that the tension
in the springs could be adjusted to keep mean length of the bellows
constant as the mean pressure was varied.
The L.V.D.T. was calibrated by displacing the core by a known
amount, and observing the output on an oscilloscope. Hence, in later
5
LVDT --Hr-
Support---
Pressure transducer
l r,---------------------~ r '
.-...... J L-
Shaker table
LVDT Amplifier
Charge Amplifier
---H~3 equally spaced springs
Thermocouple bridge
Horizontal trace
Vertical trace
Os ci 11 os cope
Fig. 1 Experimental Test Mechanical arrangement and
Instrwnentation
6
tests the voltage recorded on the oscilloscope could be converted
into stroke in inches from the calibration graph. The pressure
transducer was operated with a sensitivity of 1 picocoulomb per psi,
and the signal was displayed on the oscilloscope through the charge
amplifier. Since the transducer and charge amplifier gains were
known, the voltage indicated by the oscilloscope could be converted
into pressure units. These values were: pressure transducer - 1
picocoulomb per psi, and charge amplifier - 50 milivolts per
picocoulomb.
With the bellows compressed to its mean length, about which
oscillations took place, the volume of the bellows was measured by
measuring the quantity of water required to fill the bellows.
Plugging all instrumentation holes except one, the bellows was then
compressed for a known stroke and the measured quantity of water
displaced gave the change in volume for a given stroke.
The areas of the pressure versus displacement plots were
measured with a planimeter. These areas were converted into units
of energy by multiplying by the appropriate calibration constants.
Measurements and calculations for determining the work per cycle
are discussed in Appendix A.
All test run conditions are tabulated in Table I. One orifice
diameter of 0.025 inch was chosen to compare with the data obtained
by T. R. Townsend, and another set of test runs with an orifice of
0.0995 inches diameter was conducted to provide a smaller time
constant so that the results would encompass a larger range of WT
on the non-dimensional plots. Capillary diameters of 0.026 inches
7
TABLE I TEST RUN CONDITIONS ---··---
Type Restriction Frequency Stroke Mean 1 Range inches Pressure
cps psi a
Plugged Dead-ended 1. 4 - 15 .07, . 1, .15 14.7 orifice
Single .025 in 1. 3 - 14 .0525, .07, .1 14.7 orifice .0995 in
.026 in Single .e, 23=2.6 in
1. 2 - 10 .0525, .07, . 1 14.7 capillary .e,23
=3.9 in
Two d12 =.023 in
I
1. 2 - 10 .045, .06, .085 19.6 '
orifices d23=.025 in I
I
d12
=d23=.026 in Two 1.22 - 10 .045, .06, .085 18.6 I
capi 11 aries .e,
23=5.2 in
9-12
=6.5 in I
I ~- ~--
00
9
were nearly equal to these for the orifices, but the lengths were
selected for practical conditions and to provide a range of time
constants. The ~/d values for the capillaries were: 100, 150,
and 200.
For each test run the pressure versus displacement trace on
the oscilloscope was photographed and the area determined with a
planimeter, as described in Appendix A. From the area the energy
dissipated per cycle and hence the equivalent linear damping coef-
ficient was calculated, other test data such as frequency, mean
pressure, and supply pressure were also recorded, and are listed in
Appendix C.
The uncertainty in the dissipated energy, E, is due to variations
in area measurement, and stroke and pressure calibration constants.
This uncertainty in E is about 2.6 percent. Deviation in the
equivalent damping coefficient, Cd, is obtained by expanding the
function cd
and details
E - ---2- and is approximately 2.72 percent. Calculations
TIWX
of uncertainties in experiments are discussed in
Appendix A-6.
l 0
III. LINEAR ANALYSIS
I 'X cross-sectional area A v = A 2 X
Fig. 2 Illustration of the System Model
The model for the pneumatic damping chamber is illustrated in
Fig. 3. An expression for the equivalent linear damping coefficient
can be formulated from the equation of state for an ideal gas,
(1)
the continuity or mass balance equation,
(2)
and the energy equation,
. Q ( 3)
The energy per unit mass of the flowing stream is defined as
These equations can be linearized for small variations in the system
parameters about mean or steady-state values. The linearization is
accomplished by expanding the functions in a Taylor's series
approximation and neglecting all but the first order terms.
For example,
P P (M, V, T)
so,
since
P P (M , V , T ) 0 0 0 0
the linear variation in P is,
RT p =- m v
MRT --- v +
v2 MRt v
In the same way the mass flow rate is, in general, a function such
as
. M12 = Ml2(Pl,P2,A12'Tl)
Hence, the linearized expression lS
oM12 oM12 oM12 oM12 tl m12 = oP
1 pl + ~ Pz + oA
12 a12 +
oT1
0 0 0 0
With the assumption of an ideal gas,
Further, it will be assumed that the heat trans fer rate is propor-
tional to the temperature difference T3
- T2
. The proportionality
factor, called Ch' is an equivalent heat transfer coefficient which
is defined as a function of the system variables (see Appendix B).
Thus,
11
( 4)
(5)
(6)
or
Since T3 is constant the linear variation in Q is
The rate of energy transferred 1n the form of work is
W = p A dX 2 dt
Therefore the linearized function (assuming the mean velocity, dX dto' is zero) will be
~ = P A dx 2 dt
12
( 7)
( 8)
(9)
Introducing the time derivative operator D = ~t and with the previous
linear approximations equations (1), (2), and (3) are linearized as
equations (9), (10), and (11).
( 10)
. -Cht2 - p20ADx = CpT20M23 + CpM230t2
(11)
An equivalent linear damping coefficient can be derived by
defining a linear force representation for Eq. (8).
(12)
Hence,
If X is a sinusoidal function such as X = X Sinwt, integration for m
a complete cycle gives
or
I2Tr/w .
Wdt = E 0
E --2 Trw X
m
In the steady-state the chamber pressure will approximate a
13
(13)
sinusoidal function with a phase delay relative to the displacement,
Integration of Eq. (8) for one cycle with this function for P2
results in
E = TrAP 2 X sin¢ mm
which can be combined with Eq. (13) to give
(13a)
Thus, the equivalent linear damping coefficient can be determined
as a function of frequency, magnitude ratio, and phase angle for
the linearized analysis.
A. Dead-Ended Chamber.
If the inlet and outlet flow restrictions shown in Fig. 2 are
closed, Eq. (4) and Eq. ( 11) reduce to
M2RT2 M2R
P2 - -v2
v2 + v- t2 2
2
( 4a)
( 11a)
* After Eq. (11a) is transformed and rearranged
AP20sX(s) - -
~ + M20Cvs
combining this function for T2 (s) with the transformed Eq. (4a) gives
the transfer function
(15)
where
(15a)
and the change in volume is related to X by v2 = Ax.
For isothermal conditions t = 0 and Eq. (4a) reduces to 2
p 2 (s)
X(s) =
This result can also be obtained from Eq. (15) by letting Ch = oo.
As another special case consider an adiabatic process where . Q = 0. Then,
This result can also be obtained from Eq. (15) by letting ~ = 0. p 2 (s)
Substituting X(s) from Eq. (15) into Eq. (13a) and letting
s = jw, the equivalent damping coefficient is
*
(16)
(17)
14
Throughout the text the Laplace transformed time variable denoted by lower case symbols such asp, t, etc., will be denoted by capital symbols such as P(s), T(s), etc.
15
2 A p20A v1 +
cd (T1w)
sincj> = w v2o
vl +
( 18)
(T2
w) 2
where ¢ 0 -1 -1 -180 + tan T1W tan T2W and
T W - T2w sin¢ = 1
v1 + 2
-vl + 2 (T 1 w) (T2w)
Thus,
(18a)
In the particular case of a dead-ended chamber the reaction of
the air inside the bellows during compression and expansion is
similar to that of a mechanical spring and is sometimes referred to
in terms of the pneumatic spring constant,
k = p
(19)
The damping obtained in this case is a result of the heat transfer
between the air inside the bellows and the bellows' walls. The
defined heat transfer coefficient, Ch' is related to the damping
coefficient cd through T2 in Eq. (18a).
From Fig. 3, the graph of Cd versus frequency, it is observed
that cd is approximately a function of 1/w.
Since the heat transfer is primarily a function of the flow
it was assumed that the calculated values of Ch could be correlated
as a function of an equivalent bellows chamber Reynold's number.
This analysis is outlined in detail in Appendix C. For the
~ •ri ........... u Q)
U1 I
.D ,....,
+-> ~ Q)
·ri u
•ri 4-< 4-< Q) 0 u 01)
~ ·ri
g-(1j
'"0
+-> ~ Q)
....... (1j
> ·ri
;:::l cr'
U.l
0
. 35
. 3
.25
. 2
. 15
. 1
.OS
0 20 40 60 80 Frequency rad/sec
Fig. 3 The equivalent damping coefficient as a fl.IDction of frequency
16
100
operating range of 10 to 70 rad/sec, the analysis showed that ~
for this experiment and also for Townsend's [2] experiments could
be described extremely well by the function,
where
R = e
17
(20)
It was observed that the value of Cd was not a strong function of Ch.
B. Single Orifice Restriction.
With the inlet area A12set equal to zero in Fig. 2, and for a
small pressure difference, the weight flow formula for A23 is
(21)
Let P2 = P2m + p 2 where p 2 = ~pm sin wt, and P2m = P3 in the steady
state. Integrating the mass flow rate from 0 to TT/2w gives an
expression for the total mass change in one quarter cycle.
(sinwt) l/2 d(wt) (21a)
= 1.19 8
Eq. (21) can be linearized for small variations 1n P2 as
(2lb)
where CA is a constant. Comparing Eq. (21a) and Eq. (21b),
CA = 1.198. The time derivative of Eq. (4) yields
Dm2 " M:zo {
combining Eq. (21b) and Eq. (4b).
1. 198 -v 2gP 3
~ A23 P2 Dt2
M20 v;:p = T2
Equation (11) is reduced to
and the transform of Eq. (llb) becomes
Substituting this into Eq. (22) gives
v llPm T --
0 p2
s + v llPm
T --0 p2
where
L x
0
( s. 'oV ::m ~0 52]
T = 0
v2 -v;;_ 1.198 lf 2gRT2 A23 L
To make Eq. (23) non-dimensional assume
X n = L
18
(4b)
(22)
(llb)
(23)
Also in the steady-state,
where
0 = 0 sin(wt + ~) and n 0
t-P m
0 = and or;-Eq. (23) is thus reduced to
19
n0
sinwt
yRM2
(y-1) ( "'\ ro;- RM2 ) 2l T0 V ~ (y- 1) s J (23a)
From this transfer function the magnitude ratio is obtained from the
equation,
[ I WToCh )2 ToRM2 2] 2
+ w 4 ( l ( ~: ) (y-1)
0 3/2
[ c~ ( yRM2
w)2 J + ( 2 Ch T 0
RM2 ) ( ) +
0 0 0
+ ( y-1) no no
[ 2 ( yRM2 To l 2
w4] (24) {~Tow} + (y-1) = 0
The solution of Eq. (24) is presented graphically in Fig. 4. In
addition to the magnitude ratio the phase angles are calculated
from Eq. (23a) and illustrated in Fig. 5. Therefore, the equivalent
damping coefficient, Cd, can now be calculated from Eq. (13a) as
(13b)
10
Ojo b !="
(!.)
~ 0 !--<
.j-.l U)
0 .j-.l
C) 1 !--< ;:i U) U) (!.)
!--< 0.,
4-1 0
0 ·ri .j-.l ro !--<
(!.)
'"c:l ;:i
.j-.l
•ri
~ ro s
'"c:l (!.)
. 1 N •ri ....... ro ~ 0
·ri U)
~ C)
s •ri '"c:l
I ~ 0 z
.01 . 1
Eqn (2 3a)
Cb = 1
Cb =
ch =
1
WT 0
Eqn (25c)
Cb = ro Eqn (25 a)
Fig. 4 Non-dimensional log-log plot of the magnitude
ratio I~: I versus wT0
20
10
-90
-100
-120
-&
-140
-160
-180 . 1
Eqn (23a)
= 1.0
::: 10.
= 20
Eqn (25c) Ch = 0
1 10
WTO ~~0~ 0
Fig. 5 Phase angle ¢ versus non-dimensional term wT0
~~0 ~ 0
N ;....->
The heat transfer effects were predicted from Eq. (20). For
each value of w the value of Ch was determined and with the solution
of Figs. 4 and 5, the damping coefficient Cd was found. These can
be non-dimensionalized by Cd/k T where k is defined in Eq. (19) p 0 p
and T in Eq. (23). The calculated and experimentally derived 0
function of Cd/k T versus wT are illustrated in Fig. 6. It is p 0 0
observed that the inclusion of the heat transfer effect allows an
accurate prediction of the damping coefficient.
For the special case of an isothermal process, t 2 = 0, and
Eq. (22), after introducing the non-dimensional quantities, becomes,
22
a (s) = n
(25)
This result is also obtained from Eq. (23a) by letting Ch = oo. Thus
for the steady-state the magnitude ratio for an isothermal system is
I ~ o I = -v...!,__l_+ _4_( w_T_o_)-::4=--__ 1
o 2(wT )2
0
and the phase angle is
sin<t> = 1 (25a)
V 1 +(~wTJ The equivalent damping coefficient can then be obtained from Eq.
( 13b).
As a second special case, for an adiabatic process, Q = 0 and
1.
.1
.01
.001 .1
Fig. 6
Eqn (25a)
Eqn
1 WT
0
X Townsend's Data
a Present Data
(23a)
10
Non-dimensional log-log plot of Cd/k T versus wT p 0 0
for a single orifice restriction
23
100
24
Eq. (llb) is reduced to
After introducing the non-dimensional variables into Eq. (22), the
transfer function of Eq. (25b) is obtained.
a (s) n (2Sb)
This result could also be obtained from Eq. (23a) by letting Ch 0.
Therefore the magnitude ratio for the adiabatic system is
1~:1 and the phase angle is
sin¢ =
4 2 4(wT ) y 0
2 2 (wT )
0
1
2 - y
(2Sc)
so that the equivalent damping coefficient Cd' can be obtained from
Eq. (13b).
For lower frequencies it may be assumed that the process would
approach the isothermal case and for higher frequencies it would
tend towards an adiabatic process. However, the experimental data
indicate a larger value of Cd than predicted by the adiabatic
equation at higher frequencies. This is a result of the heat
transfer between the walls of the bellows and air. It is noted that
good agreement was achieved between the experimentally measured
damping coefficient and those calculated using the equivalent heat
transfer coefficient. For low frequencies (Fig. 4) the non-
dimensionalized magnitude ratio of pressure to stroke considering
the heat transfer coefficient converge with those of the isothermal
and adiabatic processes. While for higher frequencies (above
wT0 = 30) the curves for various Ch become asymtotic to that of the
adiabatic process.
The phase lag varies from -90° at low frequencies to -180° at
25
high frequencies as shown in Fig. 5. Ch has a greater effect on the
phase relationship than on the magnitude ratio at the higher frequen-
cies. It could be assumed that the isothermal assumption for
w < 0.4/T is valid, however, for the range of experimental data the 0
adiabatic assumption to wT of 30 is not correct. At very large 0
values of wT0
the phase angle for all ~ tends to -180° so that the
adiabatic assumption would be realized. Since the equivalent damping
coefficient is dependent on the magnitude of stroke and the frequency,
a damper would have its greatest effect if operated near the break
point frequency, w = 1/T (Fig. 6). 0
The equivalent heat transfer coefficient, ~' used in this single
orifice analysis was derived from the dead-ended chamber tests, or
Eq. (20). In general, Ch may be some function of the size of the
restriction and the bellows' geometry since the heat transfer is a
strong function of the flow phenomenon. However, for small restric-
tions the values of ~ calculated for dead-ended chamber appear to
give a good estimate of the heat transfer. It is obvious as the
26
orifice size (restriction size) is increased, the pressure difference
and the phase lag would become negligible and hence, the damping and
Ch would be reduced. Fortunately, for practical cases the dead-ended
chamber provides a good estimate of the heat transfer. Also, the
value of Cd is not a strong function of~ as noted on p. 17.
C. Single Capillary Restriction.
For this series of tests the inlet restriction in Fig. 2, denoted
as A12
, was closed. A capillary tube with diameter d 2 3 and length Q,
2 3
was installed in place of the outlet orifice restriction A23 . The
flow rate for a capillary, assuming a fully developed laminar flow is
given by
*
(26)
where
~2 = ~0 c ) ( T )3/2
: c~ 5 ~9 ~ 0 = 2.58 x 10-9 lb-sec/in2 is the viscosity of a1r at 519 °R.
c1
205 °R and c2
= 0.001433 in/lb-sec. Linearizing Eq. (26) and
neglecting variations in ~.
(26a)
where
*Reference 2 p. 44
Combining Eq. (4b) and (26a),
The transform of Eq. (llb) gives
and in combination with the transform of Eq. (27) the transfer
function relating pressure and displacement becomes
P2 (s) - -X(s)
[ where
p20 h c y [ C T --s +
-L- RM20 (y-1)
ch ch y --+
RM20 + I RM20 ( y-1)
T = c
TCS2]
;c I T S + c
T 5 2] c
( y-1)
27
(2 7)
(2 8)
For the limiting isothermal condition, t 2 = 0, and the transform
of Eq. (2 7) reduces to
= (29)
This result is also obtained from Eq. (28) by letting Ch = 00
. In the opposite extreme, the adiabatic condition, Q = 0 and the
transform of Eq. (llb) is
Combining this Eq. and Eq. (27)
1.
. 1
. 01
Fig. 7
Eqn (30)
WT c
'e ' \ Eqn (28) ,/
\
\~ ~
\
\
Non-dimensional log-log plot of Cd/k T versus wT p c c for a single capillary restriction
28
29
p20 ""L (TCS)
T S (30)
(1 + ~) y
which can also be obtained from Eq. (28) by letting Cb = 0.
From Eqs. (2 8), (29) and ( 30) I :zml and sin¢ can be obtained m
as functions of frequency. The equivalent damping coefficient Cd
can then be calculated from Eq. (13a).
Using the equivalent heat transfer coefficient, as obtained
from Eq. (20), the calculated curve for damping coefficient agrees
closely with the experimental data (Fig. 7). At low frequencies
(w < _!) the damping coefficient is constant and is equal to k T Tc p c
At these low frequencies the process could be assumed to be isother-
mal. From Fig. 7 it is apparent that for higher frequencies
(3 < wT < 40) the damping coefficient can be much more accurately c
calculated by considering the equivalent heat transfer coefficient.
D. Two Orifices in Series with Mean Flow.
With reference to Fig. 2, A12 is the inlet orifice area and
A23
is the outlet orifice area. The expression for mass flow rate
through the inlet orifice is given by
*
where N12 factor is defined as
*Reference [3~, p 20. Values o~ N12 , K12 , K23 for different pressure rat1os are tabulated 1n appenu1x.
(31)
30
[
(P /P ) 2/y- (P /P )(y+l)/yj~ 2 1 2 1 =
y - 1 2 (y+1)/(y-1
y y + 1
and factor K is given by
K = [ ( y:l) (y+l)/(y-1)] ~
yg R
For air K = 0.5318 -v-o;.; sec .
Eq. (31) is linearized and is
. [ pl p2 1 tl ] m12 = M120 (1 + K12) -- ·K -- 2 Tl p2 23 p2
where
K = (y-1)/y 1 12- (P /P )(y-1)/y- 1 y
1 2
Combining the linearized flow equation with Eq. (4b) yields,
M20 (
Dp2 Dv2 Dt 2 l . ( - (1 + K12 + K23) ~~) -- + V20 - T20 = M230 p2 +
p20 p20 2 T2
. (32)
M120 = M230 in steady state
Substituting the expression for linear mass flow rate into Eq. (11)
gives
T 10 l P2 (l+K23
) + -- K ---T20 12 p20
• yR V20 AP20 + M230 2(y-1) t2 + (y-1) Dp2 + (y-1) Dx (1lc)
and the transform of Eq. (llc) is
v20
sP2
(s)
+ (y-1)
[ S. + M23o
AP20
sX(s) + -"7""( y---=1...,.-)-
yR J 2(y-1)
J
Combining the above equation with Eq. (32)~
where
T = 00
1
...,....-!y~ 52] (y-1)
For the special case of isothermal conditions, t 2 = 0 and
Eq. (32) is reduced to
and the transfer function is
p20
L ( T S)
00
(1+T s) 00
This result is also obtained from Eq. (33) by letting Ch = oo.
For the other special case of an adiabatic process, Q = 0 and
Ch = 0 in Eq. (11c). Then combining it with Eq. (32)~
31
(33)
(32a)
(34)
32
2 2] T S 00
(35)
This result can also be obtained from Eq. (33) by letting Ch = 0.
For this case with a mean flow of air in one direction, the
value of the heat transfer coefficient Ch must include the effect of
the constant flow as well as the sinusoidal imposed flow. Ch in
this case can be estimated from Eq. (B-4) (Appendix B).
~ = 1. 2 2 5 x 10- 5 Lk [ ( R e) 0 . 8 7 5 + 2 R (Ref) 0 . 8 7 5 ] (B-4)
where
and M23 is the steady state flow rate through the outlet restriction.
This analysis is outlined in detail in Appendix B.
The procedure for calculating cd is identical to that followed
for the single orifice and single capillary cases. In brief, the
magnitude ratio and phase angle as a function of frequency are
calculated from Eq. (33). The particular Ch value having been
derived from Eq. (B-4). The value of Cd is obtained from Eq. (13a).
The limiting conditions of adiabatic and isothermal processes are
realized by setting Ch = 0 and ~ = oo respectively. These values for
Cd are non-dimensionalized by dividing by the product k T and graphed p 00
versus WT in Fig. 8. The improved prediction with the addition 00
of the heat transfer coefficient is shown by the close agreement
of the experimental and calculated values. At wT of 70, for 00
example, the assumption of an adiabatic process would be in error
'"d u
1.0
. 1
(\
' ' ' \ \ Eq. (33) ~ -o,/ 0
' t---?01 p..,
~ ~
\ \ ~
~
'\ .001 '
.0001 ~--~---L-L~~~~----~-L~-L~~U-----~-L~-L~~ . 1
Fig. 8
1 WT
00
10
cd
Non-dimensional log-log plot of versus wT k T 00
p 00
for two orifices in series with mean flow
100
33
34
by a factor of 77 percent in predicting cd.
E. Two Capillaries in Series with Mean Flow.
In Fig. 2 both restrictions are capillary tubes. Flow rate
through the outlet capillary is given by Eq. (26). Linearizing the
weight flow formula, Eq. (10) is
(36)
where the partial derivatives are
(36a)
(36b)
3 1 -+ 2 1 + T l/C
1
(36c)
Neglecting variations in p1
, p3
, and t 1 and combining equations
( 36) and ( 4b)
( 37)
The transform of Eq. (37) is
For capillaries with mean
-Cht2 - P2ADx
flow, Eq. (11) reduces to .
oM23 ) CpT20 ( oM23
= ~p2 + ~t2 2 2
c v
+ R v20Dp2
and the transform of Eq. (lld) is solved for T2 (s)
-'-. [ C + C M• + C T 0
M2 3 ] h p 23 p 20 oT2
combining this equation with Eq. (37a)
P 2 (s)
X(s) =
2 P ( y-1) A" T )
L20 [ ( YTcc + ch RM20 cc s
2 2 ] + ( yA"T ) S .;-cc
[ Y + _c_h_C_Y_-_l_)_A_"_'_cc+
RM20
2
I ~ (y-l)A11 T cc +
RM20
+ YT cc!'-" + B'.'T cc) s +
y(l-B")T cc
35
(37a)
(lld)
(38)
where
and
and
T cc =
A"
2 2P20 2 2
pl0-P20
2 2P 20.
for simplification,
Considering the extreme case of isothermal process, t 2 = 0 in
Eq. (37) and taking the transform of that equation
=
p20 -
1 (T s) cc
(1 + T s) cc
This result can also be obtained from Eq. (38) by letting Ch = oo
For the other extreme case of an adiabatic process, Q = 0 and
ch = 0 in Eq. (lld). Combining that equation and Eq. (37a)
p 2 (s) Pzo [ yA"T s2 ] [ y + X(s)
- -- YT s • L cc cc
( T y(A"+l-B") + B" ) 2 2] s + A"T s cc cc
This result can also be obtained from Eq. (38) by letting Ch = 0.
As mentioned on p. 32 for one case of two orifices with mean
flow,~ with mean flow can be estimated from Eq. (B-4), where
23 . Ref = ~ and M
23 will be the steady state mass flow rate through
m outlet capillary.
36
( 39)
c 40)
the
37
After substituting the value of Ch for a corresponding frequency
ln Eq. (38) the magnitude ratio of pressure to stroke and the phase
angle can be calculated. With these values Cd is then obtained from
Eq. (13a). Similarly, from Eqs. (39) and (40), Cd is calculated for
the extreme cases from Eq. (13a). Non-dimensionalized quantities of
Cd/K T versus WT obtained experimentally and calculated are p cc cc
illustrated in Fig. 9.
At lower frequencies (wT < 1) the damping coefficient becomes cc
constant and the process could be assumed to be isothermal. The
plot (Fig. 9) is very similar to the multiple orifice case (Fig. 8)
with the exception of the magnitude (and definition) of T . Again cc
at the higher frequencies (wT > 5) the error in calculating Cd by cc
the adiabatic assumption becomes larger up to the experimental range
of the frequencies.
1.
. 1
.01
.001 . 1
Eq. (39)
Eq. (40)
WT cc
Fig. 9 Non-dimensional log-log plot of Cd/k T versus wT for two
p cc cc
(38)
'
capillaries in series with mean flow
38
\ \
\
39
IV DISCUSSION
If the polytropic constant n is considered for relating pressure
n-1 T and temperature (t = -n- p p), the transfer function for two orifices
with mean flow will be
where
P 2 (s)
X(s)
T = 00
=
p20 -L- T s
00
T 00
+ -- s n
with n = 1.0, the isothermal condition, Eq. (41) reduces to
p20
( 41)
= -L- T s
00
(1 +T S) ( 41a) 00
which is identical to Eq. (34). The same considerations for the
limiting case of an adiabatic process (n = y) or ~ = 0, results in
the magnitude ratio, phase angle, and damping coefficient, calculated
from Eqs. (35) and (41). The values of Cd are tabulated in TABLE II,
for comparison of the two methods. Although the equations do not
compare exactly for the adiabatic case the results in TABLE II show
that the difference in calculating Cd from Eqs. (35) or (41) becomes
negligible for larger WT . Magnitude ratio of pressure to stroke, 00
phase angle and the equivalent damping coefficient, Cd, as a function
of frequency for different values of n could be observed in Fig. 10.
The physical interpretation of n, as applies for polytropic processes
in thermodynamic considerations, does not have the same meaning for
WT 00
rad/sec
.16
.8
1.6
8.0
16.
32.
80.
pzlx
TABLE II COMPARISON OF CALCULATED VALUES FOR THE ADIABATIC PROCESS ASSUMPTION
From Eq . ( 35) From Eq. (41)
<P cd pzlx <P
1.031 -97.4 51.08 1.088 -96.8
4.41 -119.4 38.37 4.47 -120.8
6.76 -137.82 22.67 7.04 -139.91
9.05 -170.65 1.56 9.08 -170.46
9.15 -175.8 0.415 9.19 -175.
9.2 -178.4 0.105 9.2 -177.6
9.2 -178.96 0.016 9.2 -179.04
cd
53.3
40.2
22.76
1.5
0. 391
0.097
0.0156
~ 0
.._, 60 ~
C!)
•rl u
•rl 40 4-1 4-1
C!) 2.5 .._, 0 ~ u 20 C!) n = 1.0
- bJ) ro ~ >·rl ·rl p.. 0
30 g.~ P-l"U
.1 10 Frequency rad/sec
Fig. 10 Magnitude ratio, phase angle and equivalent damping coefficient, Cd' as
a function of frequency for different values of n
41
so
42
the processes considered in the present analysis. A much better
prediction and clearer understanding of the physics of the present
processes can be obtained with the consideration of the heat transfer
through Ch. Furthermore, on page 17, it was shown that the value of
Cd is not strongly influenced by ~ whereas it is a strong function
of n. Therefore the estimation of the exact value for ~ is not as
critical as the assumption of the correct values for n.
For low frequencies, an isothermal process could be assumed up
to WT = 0.7 for a single capillary restriction. At non-dimensional c
frequencies (wT) greater than 70 an adiabatic process with Q = 0
could be assumed. The range for assuming isothermal process for
the mean through-flow cases with two orifices and two capillaries
in series would be wT < 1.6 and wT < 1 respectively. The adiabatic 00 cc
assumption for two orifices could be assumed for wT > 110 and for 00
two capillaries it would be wT > 40. At the intermediate frequencies cc
(10 to 70 rad/sec) which are the most practical for pneumatics, the
heat transfer coefficient should be estimated for accurate predictions.
As illustrated in Figs. 6, 7, 8, and 9, the assumption of Q = 0
provides a conservative estimate for Cd.
In Fig. 6, the damping coefficient for the single orifice is a
1 maximum at w = - and
T
. f 1 1s less or w > -T
0
1 Thus the value or w <
0
of T for the orifice 0
should be selected
maximum damping at a frequency of 1
w =T
T 0
to obtain the desired
On the other hand a
0 1 damping coefficient for w < capillary (Fig. 7) has a constant T
. 1 and has about one-half the max1mum Cd at w = ;-
c Therefore a
c
capillary would give much greater damping than an orifice for low
frequencies. At frequencies greater than ! the damping coefficient T
for either orifices or capillaries in series decrease approximately
as the square of the frequency.
The results presented in chapter III can be used to estimate
Cd in the design of pneumatic damping chambers. Suggested calcu
lated steps for determining Cd are: 1. For the cases of a single
orifice or a single capillary with no mean flow, the value of Ch is
estimated from Eq. (20). With mean flow Ch is estimated from
Eq. (B-4) where M23
is the steady-state flow through the outlet
43
restriction. 2. From the system dimensions and design specifications
the time constant T can be found from Eqs. (23), (28), (33), or (38).
3. Magnitude ratio of pressure to stroke and phase angle can be
calculated from the transfer functions. These are Eq. (23a) for a
single orifice, Eq. (28) for a single capillary, Eq. (33) for two
orifices in series with the mean through-flow and Eq. (38) for two
capillaries in series with mean through-flow. 4. The equivalent
damping coefficient Cd is then obtained from Eq. (13a).
As an example of the application of the result reported herein,
the case of a pneumatic dashpot as shown in Fig. 11 is considered.
Suppose it is desired to have c = 0.2 for the example. The equation
of motion with damping coefficient Cd will be
= -{!5!_ W/g
= ""'\ /31.6 X 386 v 25
= 22.05 rad/sec
orifice-
w = 25 lbs
D 2 inches m
L 4 inches
p2 = 14.7 psi a
k = 20 lb/in
A = 3.141 sq
v2 = 12.55 cu
w
k 1 L
orifice~ l (a) (b)
Fig. 11 (a) Piston cylinder arrangement (b) Bellows chamber arrangement
in
in
P A2 k = 2 = 11.6 lb/in
p --v;-T2 = 532° R
44
Maximwn stroke X = 0.2 inches.
ch
R e
f.!m at 532°R =
k =
Therefore
R = e
ch =
=
The damping coefficient
cd =
c = d
Solving forT from Eq. (23a), 0
=
To obtain heat
P2D X w mm T2f.!m
2.625 10-9 lb X
sq
0. 0149 7
92.5 X 10 6
1. 225 X 10-5 kL
6.94
2?;;mw = n 0.5 72
1::1 p2 . A L s1n¢ w
n
T 0. 0815 0
transfer coefficient
sec in
(R ) . 875 e
T = 0
------~v=2==~---~ 1.198 V 2gRT2 A23
Therefore, A23 = .001781 sq. in. and diameter of the square edged
orifice is d23 = 0.0525 inches.
For designing a capillary for the same conditions,
45
46
From Eq. (28),
T = 0. 2595 c
V2L23 T =
c 4 )JOTO
2P 2
RT 2c
2 (10d
23)
ll2T2
llo To 0.946
ll2 T 2 =
Therefore,
L23 2 79.5
4 ( 10d2 3)
For L23 = 2 inches, d23 = 0.0343 inches.
47
V. CONCLUSION
It can be concluded that an improved prediction of the damping
coefficient can be achieved by considering the heat transfer effects
rather than assuming a polytropic constant n for relating pressure
to temperature. Although the usual assumption of an isothermal
process at low frequencies is valid, the assumption of an adiabatic
process for the normally encountered frequencies is not valid. In
fact the error for the adiabatic assumption compared with the analysis .
which includes an estimate of Q is of the order of 70 percent. The
steps for calculating the equivalent linear pneumatic damping
coefficient Cd for orifice or capillary restrictions with and without
mean through-flow are outlined on p. 43.
Future work should include investigation of the equivalent
heat transfer coefficient ch for the frequencies beyond the present
test range of 10 rad/sec to 70 rad/sec. Also as noted on p. 25
the functional behaviour of Ch with the size and geometry of the
restriction and the bellows chamber should be studied further. The
variation of the number, the dimensions and the types of restrictions
simultaneously installed on the same dashpot or bellows might give
maximum damping peaks for a range of frequencies. Also some control
could possibly be employed to vary the restriction geometries or
openings. Other ideas which should be considered are the use of
a fixed or variable plenum chamber, or simply filling the bellows
with a porous and flexible material.
APPENDIX A
SYSTEM DIMENSIONS AND INSTRUMENTATION CALIBRATIONS
A.1 Pneumatic Dashpot Dimensions
Internal mean volume of the bellows
v2 = 243.5 ml
= 14.9 in3
Compression stroke of 0.2" displaces 16.4 ml of water
16.4 ml - 1.0 in 3
Therefore, cross-sectional area,
1 A= 0 . 2 = 5.0 sq. in.
A.2 Displacement Transducer Calibration
Sensitivity: 50/1000
Oscilloscope horizontal trace: 1 volt/em
LVDT calibration curves are shown in Fig. 12.
A.3 Pressure Transducer Coefficients
Sensitivity: 1 picocoulomb/psi
Charge amplifier gain: 50 milivolts/picocoulomb
Oscilloscope vertical trace: 10 milivolts/cm
Therefore,
1 em 10 = 50- 0. 2 pcb
= 1 pc /psi
A.4 Oscilloscope Trace Area Measurement
For the horizontal trace the oscilloscope setting was 0.5
volts/em and with the LVDT gain of 28.5 volts/in., the horizontal
48
Ul ~
4
3.5
3
2.5
c; 2 >
1.5
1
. 5
0 0 .025
Fig. 12
.05
Two orifices with flow and two capillaries with flow
.075 stroke inches
cases with no mean flow
49
.1 . 125
LVDT Calibration curves
50
scale is
V 1 in 0 ·
5 em' 2 s. 5 v 1 in = 57 em
From the pressure transducer calibration of 50 mv/psi and the
vertical oscilloscope gain of 10 mv/cm the vertical scale was
~ psi/em. Thus the oscilloscope trace area in cm2
can be reduced
to equivalent energy units by the conversion factor,
1 in } ( .!_ ~} 57 em 5 em
(S in2) = _!__ lb in 57 2
em
Therefore
1 cm2 = 0.0175 in-lb
In some cases the horizontal trace of the oscilloscope had to be
set for 1 volts/em so that this area conversion factor would be multi-
plied by a factor of two.
A.S Magnitude Ratio and Phase Angle Measurements
The area of the loop is a measure of the amount of energy con-
verted to heat and the arrows in the diagram indicate the direction in
which the loop is developed. In the case illustrated in Fig. 13,
X = X sinwt m
and the resulting pressure is
p P sin(wt+<j>) m
which lags in phase with respect to stroke by the angle ¢.
At point a,
so,
P = 0 = P sin(wt+<j>) m
wt = -<j>
P = P sin(wt + ¢) m
x = X sinwt m
Fig. 13 Formation of elliptical loop and measurements of magnitude ratio and phase angle.
51
f
and
Thus
X a X= sin(- cp) m
X = X sinwt a m
X sin(- cp) m
(cp itself is negative)
The phase angle is therefore obtained from the photographs by
measuring lengths X and X a m
. -1 xa cp = s1n X
m
The magnitude ratio of pressure to stroke is given by measuring P m
and X . m
A.6 Uncertainties in Experiments
In all experiments uncertainties in results can result from
accuracy error and/or precision error. Accuracy error is detected
by a simple calibration procedure while precision error implies a
random fluctuation of the instrument reading about the true value of
the measured quantity
Uncertainty can be supposed as having a distribution of values
around the true reading so that standard statistical principles can
be applied. In the present context there is uncertainty in reading
the values from the photographs, and measuring the orifice diameter
and bellows volume, etc.
A general function,
Z = f(U,Y)
can be expanded as
52
53
Z + z = f(U + u, Y + y) c c c
where the subscript c refers to the true or average reading, u and y
are deviations of U and Y measurements from U and Y , and z is the c c
deviation of the result. If the function is continuous and has
derivatives it can be expanded in a Taylor series, using the first
two terms only.
Since
Z = f(U , Y ) c c c
the expression reduces to
z = ( ~~cl u +
y
L: z 2 ( ~~c) 2
( ~~c) ( ~) L:uy = L:u + 2 oY c y y u
+ ( ~; c l L:y2
u
Term L:uy tends to zero so that the mean deviation squared is defined
as
where
and
oz ) 2 ~ u
c y
2 z =
2 y =
L:z 2
n
2 ~
n
n
oz ) 2 8'1 y c
u
for n number of readings.
2 Dividing by Z and taking the square root gives an estimate of c
the percent deviation.
( .!__ g_)2 z ov
c c
2 u +
Uncertainty in the calculated areas from the photographs is approxi-
mated from length, L, and height, H, in the photos.
A= LH
and variation in A is
a=Lh+~H
The uncertainty in the dissipated energy, [, is due to variations in
area measurement, and stroke and pressure calibration constants.
Expanding the function as mentioned for the general case,
% deviation in E =
From the experimental values, this uncertainty in E is about 2.6
percent. Similarly the deviation in damping coefficient Cd is
obtained by expanding the function
E cd = --2
TIWX
and is approximately 2.72 percent.
54
55
APPENDIX B
DERIVATION OF EQUIVALENT HEAT TRANSFER COEFFICIENT, Ch
For the dead-ended chamber (inlet and outlet plugged) the energy
dissipated per cycle must equal the heat transferred. Thus a heat
transfer coefficient can be derived from these data. In Fig. 3 the
value of Cd is plotted versus frequency. This is related to the
equivalent heat transfer coefficient as expressed in Eqs. (6), (15),
and (20). The values of~ calculated from Eq. (20) are shown in
Fig 14 as a function of w. It was assumed that the equivalent
convection heat transfer coefficient could be related to a Reynolds
number and Prandtl number as has proved valid and useful for many
convection heat transfer processes. To test this assumption a
Reynolds number based on a mean bellows velocity (and hence air
velocity) was defined as
R = ew
Since all tests were essentially at the same mean temperature and
were conducted with air only, the property variations and Prandtl
number dependence could not be tested. From Eq. (6)
Ch = DLh (B-1)
and defining Nusselt number hD/k,
C = K Lk (R ) \jJ h o ew
(B-2)
or
log C = log (K Lk) + \jJ log R -h o ew
The data from Fig. 14 were replotted as the log ~ versus log Rew
c.f'
25~--------------------------------------------------------------------------,
20
15
10
5
... / ,.
o I 0 10
Fig. 14
two orifices with mean flow
20 30 40 so w
/ ,.
single orifice and single capillary restriction
60 70
Equivalent heat transfer coefficient, Ch' as a function of frequency w rad/sec.
80
Vl 0'
57
(Fig. 15). Although the plot indicates a possible upward curve at
the higher frequencies the limitation of the data at w > 70 rad/sec
(due to the system and shaker-table) and the experimental uncertainty
do not allow a definite decision. For the experimental range of
10 < w < 70 rad/sec a straight line function fits the data within the
experimental uncertainty. The slope of this line is ~ = 0.875 which
agrees very closely with published results for turbulent convective
heat transfer correlations. With the straight line function the
magnitude of K in Eq. (B-2) found to be 1. 225 -5 was X 10 .
0 Thus the
final equation without mean flow is
ch = 1. 225 X 10-S Lk (R ) 0. 875 ew
(20)
This result was tested against Townsend's [2] dead-ended bellows data
and agreed within 7 percent of his measured values. His bellows
dimensions were L = 1. 718 in. and v2
= 5.97 in. 3
, as compared with
L = 2.98 in. and v2 = 14.9 in.3
.
For the tests with mean flow through the bellows it was apparent
that the calculation of the heat transfer coefficient must account
for the higher mean velocity (Reynolds number). It was assumed that
the total coefficient could be thought of as composed of frequency
and mean flow components. Thus,
(B- 3)
where
M23
is the steady state flow rate through the outlet restriction.
58
1.4
1.2
1.0
. 8
r.f' 0 rl
bl)
0 ...J
.6
.4
. 2
0 7.5 7.7 7.9 8.1 8.3 8.5
Log lORe
Fig. 15 Log of equi vaient heat transfer coefficient, ch, versus log of Reynolds' Number.
Since the heat transfer is a function of Re to the 0.875 power
it seems immaterial whether or not the flow was generated by the
bellows or a fixed mean pressure difference. Thus it was assumed
~f = 0.875
and further that
Kf -5 2R - 1.225 X 10
so that
59
C = 1.225 x 10-5
Lk [ R0
·875
+ 2R R0
· 875 J ~ ew ef (B-4)
This assumed relation was tested by calculating Ch values for the
mean flow tests with orifices and capillaries. These values, which
are graphed in Fig. 14, were used in calculation of Cd for Figs. 6,
7, 8, and 9. The extremely good prediction (within the experimental
uncertainty) indicate that Eq. (B-4) can be used with confidence
over the test frequency range of 10 < W < 70 rad/sec.
60
APPENDIX C
EXPERIMENTAL RESULTS
C.1 Experimental Data Reduction
The tabulated results for all tests are listed in TABLES III,
IV, V, VI, and VII. As an example of the data reduction technique
the particular case of run No: 2 with single orifice is illustrated.
Run No: 2, TABLE IV, single orifice
Frequency: 12.55 rad/sec
Stroke: 0.07 inches
2 Area of loop - 7.485 em
E = Area x conversion factor
= 7.485 X 0.0175
. 13097 in-lb
cd E .13097 = --2- 12.55 .0049 TT X X
TTWX 0
0.6775 lb-sec = in
C.2 Time Constant Calculations
The time constant for each of the four configuration tested were
calculated. They are
Single Orifice:
T 0
= ------~v=2==~---~ 1.198 V2gRT2 A23
= ~rx: 14.9 v~
-1-.-19_8 __ \1~~2==x==3=86==x==6=3=9=.=6=x==5=3=4===4=.=0=2=x==l=0~4 2 · 98
for
Single Capillary:
p A2 K 2
p = --v:;- =
T = c
X = 0.07 0
T = 0.29 sec. 0
14. 7 X 25 14.9
K T = 7.15 p 0
25.65 lb/in
( 2.)3/2
519
where ~ lS the viscosity at 519°R and 0
0 c1 = 205 R, a constant
C2
= 1.433 X 10-4 in lb sec
For d23
= 0.026 and L23
2.6,
T = 0.437 c
For d23
= 0.026 and L23
= 3.9,
T 0.655 sec c
p A2
K = -2--- 25.65 lb/in
P v2 -
Two Orifices in Series with Mean Flow:
T = 00
61
K 0.5318 0 = R/sec
R 639.6 in/0
R
A23 = 0. 82 X ~ (0.025)2
4.02 154
in 2
= X
T2 534.0 OR
v2o 14.9 in 3 =
For pressure ratios,
Two
K p
Capillaries
T = cc
( Rt2
= v:;--K T =
p 00
with Mean
2 2P20
2 2 +
p10-P20
K12 = 1.3785
K23 = 0.9685 and N23
T = 1.16 sec 00
19.6 X 25 14.9
32.88 lb/in
52.6 lb/in
Flow:
P20v2o 2
) 4
2P20 (10d23) c2 2 2
p20-P30 1
23
~2 = f.lo ( 519 + c 1 ) I 2_)3/2
T2 + c 1 \ 519
0. 8836
~ T 2 2 0 0
~2T2 (P 20-P 30)
where c1
= 205°R and ~0 = 2.58 x 10-9 lb-sec is the viscosity of a1r in
62
123 =
d23 =
R =
Therefore
fl T 0 0
0.9506 fl2T2 =
p 10 22.7 psi a
p20 = 18.7 psi a
p30 = 16.7 psi a
p2_p2 123 2 3 0.8 1 12 = =
p2_p2 1 2
5.2 inch and 112
0.026 inch, d12 =
639.6 in/ 0 R
T = 0.562 sec cc
p A2
K 2
31.4 = --v;- -p
K T = 17.6 p cc
C.3 Tables and Typical Test Run Photographs
63
6.5 inch
0.026 inch
lb/in
No. Frequency cps
1 1.4
2 2
3 3
4 5
5 5
6 5
7 10
8 10
9 10
10 15
11 15
TABLE III EXPERIMENTAL DATA FOR DEAD-ENDED CHAMBER
Stroke p2 sin ¢ Area Energy cd inches - 2 E 1b-secjin X em
in-lb
.07 5.94 .1186 2. 3948 .0419 . 3097
.07 6.56 .1 2.323 .0407 .2105
.07 6.4 .0872 1. 839 .0321 .1109
.07 6.56 .1135 2.365 . 04138 .0856
.1 6.32 .0872 1. 828 . 06398 .06487
.15 6.3 0.941 1. 716 .1501 .0676
.07 6.85 .0958 2.124 .03717 .03845
.1 6.8 .068 1.469 .05141 .026
.15 6.67 .0889 1.599 .1399 .0315
.07 6.85 .1083 2.237 . 03914 .0269
. 1 6.8 .0962 2.119 .07416 .025
Mean pressure P2 psia
14.7
14.7
14.7
14.7
14.7
14.7
14.7
14.7
14.7
14.7
14.7
Q\ .j::.
TABLE IV EXPERIMENTAL DATA FOR SINGLE ORIFICE RESTRICTION
No. Frequency Stroke p2 sin ~ Area Energy cd cd inches 2 E cps -1b-sec/in kT X em
in-1b p 0
1 1.3 .07 5. 71 .525 9.678 .1693 1. 348 .1885
2 2 .07 6.0 .425 7.485 .1309 .6775 . 094 75
3 3.5 .07 6.96 .242 5.49 .0961 . 284 .0397
4 4 .07 6.78 .2482 5.5 .0962 .249 .0348
5 5 .07 6.57 .2185 4.58 .08015 .1657 .02317
6 7.5 .07 7.0 .163 3. 728 .0652 .09 .01258
7 10 .07 7.0 .141 3.214 .0562 .0582 . 008139
8 15 .07 7.0 .136 3.1 .0543 .0375 .00524
9 3.2 .1 2.1 .966 5.56 .1945 . 308 .568
0 6.5 .07 3.57 .79 9.678 .1693 .2695 .594
1 14 .0525 5.34 .609 6.0 .105 .1385 . 35
WT 0
2.366
3.642
6.38
7. 279
9.108
13.659
18.21
27.31
.442
. 751
1.4
d23
,025 I
.025 i
.025
.025
.025
.025
.025
.025
.0995
.0995
.0995
0\ (Jl
TABLE V EXPERIMENTAL DATA FOR A SINGLE CAPILLARY RESTRICTION
No. Frequency Stroke p2 ~ Area Energy cd cd inches 2 E cps - lb-sec/in k"T X em in-lb p c
1 1.06 .0525 4.95 45.5 6.322 .1106 1. 923 .178
2 2.15 .07 5. 714 25.9 8.065 .14128 .6792 .0628
3 4.2 .07 6.43 15.2 5.81 .1016 .2502 .0232
4 1.2 .0525 5.904 28 5.033 .0088 1. 352 .0838
5 3 .1 6.8 13 5.355 .1874 .3166 .0196
6 4 .07 7.06 11.45 6.158 .1077 .279 .0172
7 5.5 .07 7.0 7.9 3.2905 .05758 .1082 .0067
8 7 .07 7.14 8.7 3.513 .0614 .091 .0434
9 10 .07 7.0 7.2 3.226 .0564 .0583 .0036
WT c
2.9
5.9
11.5
4.94
12.34
16.45
22.6
28.8
41.2
£/ d
100
100
100
150
150
150
150
150
150
(J\ Q\
No. Fre-quency cps
1 1.1
2 1.4
3 2
4 2.5
5 3
6 4
7 5
8 7
9 10
TABLE VI EXPERIMENTAL DATA FOR TWO ORIFICES IN SERIES WITH MEAN THROUGH-FLOW
Stroke p2 sin ~ Area Energy cd cd urr p2 d12 inches 2 E 00 -lb-sec k T X em psi a inch in-1b p 00 ln
.045 7.55 .314 3.875 .0581 1.326 .0252 11.05 19.6 .023
.0525 7.61 .334 4.629 .081 1. 067 .0203 16.05 19.6 .023
.06 8.17 .25 6.1 .0915 .6477 .0123 20.1 19.6 .023
.085 8.35 .225 5.39 .1617 .4538 .0086 25.12 19.6 .023
.07 8.96 .2192 6. 395 .1119 . 386 .00734 30.1 19.6 .023
.06 9.0 .1875 4.58 .0687 .2420 .0046 40.16 19.6 .023
.07 9.15 .1565 4.639 .08119 .168 .0032 50.2 19.6 .023
.06 9.15 .1375 3.42 .0513 .1033 .00196 70.4 19.6 .023
.06 9.15 .1125 3.355 .05032 .07086 .00135 100.5 19.6 .023
d23 inch
.025
.025
.025
.025
.025
.025
.025
.025
.025
0\ -.._)
TABLE VII EXPERIMENTAL DATA FOR TWO CAPILLARIES IN SERIES WITH MEAN THROUGH-FLOW
No. Frequency Stroke P2 ¢ Area Energy cd cd W1
inches 2 E cc cps -
X em 1b-sec/in k 1 in-1b p cc
1 1. 22 .045 6.67 18.7 5.242 .0915 1.862 .106 4.31
2 1. 54 .045 7.21 16.1 4.46 .078 1. 26 .0716 5.44
3 1.95 .06 6.98 16 8.941 .1562 1.13 .0641 6.88
4 3.0 .06 7.14 13.6 6.325 .1107 .519 .0295 10.58
5 4.28 .06 7.428 9.25 5.173 .0879 . 289 .0164 15.09
6 7 .085 8.6 7.2 2.13 .1532 .154 .00875 24.65
7 10 .06 8. 71 5.85 4. 211 .0703 .099 .00562 35.3
£2/d23
200
200
200
200
200
200
200
0\ 00
Fig . 16 Photograph of test run No: 2 of TABLE III
Fig. 17 Photograph of test run No: 8 of TABLE III
69
Fig. 18
Fig. 19
Photograph of test run No: of TABLE IV
Photograph of test run No: of TABLE IV
70
2
5
Fig. 20 Photograph of test run No: 9 of TABLE IV
Fig. 21 Photograph of test run No: 10 of TABLE IV
71
Fig. 22 Photograph of test run No: 2 of TABLE V
Fig. 23 Photograph of test run No: 3 of TABLE V
72
Fig. 24 Photograph of test run No: 5 of TABLE V
Fig. 25 Photograph of test run No: 7 of TABLE V
73
Fig. 26 Photograph of test run No: 3 of TABLE VI
Fig. 27 Photograph of test run No: 8 of TABLE VI
74
Fig . 28 Photograph of test run No: 3 of TABLE VII
Fig. 29 Photograph of test run No: 7 of TABLE VII
75
BIBLIOGRAPHY
1. R. L. Peskin and E. Marti zen, "A Study of the Response of a
Small Porous Chamber to Forced Oscillations", A.S.M.E. Trans.
Series D ~' 25-32 (March 1966).
2. R. W. Townsend, "Damping Characteristics of a Pneumatic
Dashpot", M.S. Thesis, Arizona State University, Arizona,
1965.
3. B. W. Anderson, The Analysis and Design of Pneumatic Systems
(John Wiley and Sons, Inc., New York (1967)).
4. Hilbert Schenck, Jr., Theories of Engineering Experimentation
(McGraw-Hi 11 Book Co., Inc., New York (1961)) .
76
VITA
Nathalal Gordhanbhai Patel was born on July 28, 1942 in Tabora,
Tanzania. He received his primary and secondary education in Tabora,
Tanzania. In 1961 he obtained a School Certificate, Cambridge
University, and went to India for college study. He received a
Bachelor of Engineering degree in Mechanical Engineering from the
Maharaja Sayajirao University of Baroda, India in June, 1966. After
graduation he worked for twenty-one months with the Tanzania Electric
Supply Company, Tanzania, as Station Engineer.
He has been enrolled in the Graduate School of the University of
Missouri - Rolla since September, 1968.
77
